What remains undetermined when only the derivative is given?

The added constant. Different constants give the same derivative.

Which entire factor becomes dt here?

2xdx Differentiating x squared supplies the factor twice x.

Which missing factor prevents direct substitution of sine into the power rule here?

cos x Cosine is the derivative of sine.

What coefficient outside inverse tangent corrects the first derivative shown?

frac12 The uncorrected derivative is twice the required expression.

What operation joins the two expressions inside the combined logarithm?

Division. Multiplication would give a sum of logarithms.

Which sign belongs before the remaining integral in the last line?

Minus. Subtracting removes the extra term created by the product rule.

What is this exact integral, including any constant needed?

1, with no arbitrary constant. The triangle has a fixed base and height, so its area is fixed.

The two equal integrals together give one. What is each integral?

One half. One is the combined total of two equal areas.

Teach Integrals in eight steps.

For each idea: show it, ask before you tell, then say one line.

Gather paper and a pencil.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

Integrating a car’s speed over time gives the distance travelled.
Integrating a tap’s flow rate over time gives the volume collected.
An electricity bill depends on energy accumulated as appliances use power.

What to teach

How to teach it

Reverse differentiation

Differentiating x squared gives twice x.

$\frac{d}{dx}(x^2+C)=2x$

1. Integration as an Inverse Process of Differentiation

Find functions whose derivative matches what you know.

Show: Write these lines. Say: “Here, x varies and C is any constant.” First write that differentiating x squared gives twice x. Point to the matching derivatives.
Ask: What should we write for the integral of twice x?
Say: Any constant disappears on differentiation, so the integral is x squared plus C.
Used in: Integrating speed gives distance travelled, helping measure a trip.

If your child says…“There is only one possible answer.”

Ask: What information about the constant survives differentiation?
What’s true: Adding any constant leaves the derivative unchanged.

Call x squared t.

$dt=2x\,dx$

$\int 2x\cos(x^2)\,dx=\int\cos t\,dt$

$\int 2x\cos(x^2)\,dx=\sin t+C$

2. Integration by Substitution

Rename the inside expression and change the differential with it.

Show: Write these lines. Say: “Call x squared t; twice x times dx becomes dt.”
Ask: What does the last line become back in x?
Say: The answer is sine of x squared plus C.
Used in: Integrating changing water flow gives the amount collected, helping plan tank filling.

If your child says…“I can just replace dx with dt.”

Ask: What happened to twice x when you changed variables?
What’s true: The differential changes according to the derivative of the substitution.

Choose a method

$\cos 2x=1-2\sin^2x$

$\sin^2x=\frac{1-\cos 2x}{2}$

$\int\sin^2x\,dx=\frac12\int(1-\cos 2x)\,dx$

$\int\sin^2x\,dx=\frac12\left(x-\frac{\sin 2x}{2}\right)+C$

3. Integration Using Trigonometric Identities

Rewrite trigonometric expressions into forms you can integrate.

Show: Write these lines. Say: “The identity replaces sine squared with simpler terms.”
Ask: What does the last line become after multiplying out the half?
Say: Half of x minus a quarter of sine of twice x, plus C.
Used in: Integrating squared alternating current helps calculate the heat produced in a resistor.

If your child says…“Sine squared uses the ordinary power rule directly.”

Ask: Where is the cosine factor needed for that substitution?
What’s true: Using the power rule on sine requires its derivative, cosine, alongside it.

$\frac{d}{dx}\tan^{-1}\left(\frac{x}{2}\right)=\frac{2}{x^2+4}$

$\frac{d}{dx}\left[\frac12\tan^{-1}\left(\frac{x}{2}\right)\right]=\frac{1}{x^2+4}$

4. Integrals of Some Particular Functions

Match a standard form, including its scale, signs and roots.

Show: Write these lines. Say: “Inverse tangent returns an angle; the half corrects its derivative.”
Ask: Which integral follows from the second line?
Say: Half the inverse tangent of half x, plus C.
Used in: Integrating internal cross-sectional areas along a container’s length gives its capacity.

If your child says…“Four is the number that divides x inside inverse tangent.”

Ask: Which number squared gives four?
What’s true: The denominator contains a squared scale, so four corresponds to a scale of two.

$1=A(x+2)+B(x+1)$

$\frac{1}{(x+1)(x+2)}=\frac{1}{x+1}-\frac{1}{x+2}$

Integrate the two fractions separately.

$\int\frac{dx}{(x+1)(x+2)}=\log|x+1|-\log|x+2|+C$

5. Integration by Partial Fractions

Separate a rational expression into simpler fractions before integrating.

Show: Write these lines. Say: “A and B are constants; log means natural logarithm.” Substitute −1, then −2, to find the constants.
Ask: What single logarithm replaces this difference?
Say: Log of the absolute quotient, x plus one over x plus two, plus C.
Used in: Partial fractions simplify flow calculations to estimate draining times.

If your child says…“Subtracting logarithms means multiplying what is inside.”

Ask: Which logarithm rule turns division into subtraction?
What’s true: A difference of logarithms combines through division.

$\frac{d}{dx}(xe^x)=xe^x+e^x$

$xe^x=\frac{d}{dx}(xe^x)-e^x$

$\int xe^x\,dx=xe^x-\int e^x\,dx$

6. Integration by Parts

Reverse the product rule by subtracting an extra integral.

Show: Write these lines. Say: “The exponential, e to the power x, is its own derivative.”
Ask: What remains after integrating the final exponential?
Say: Subtract the exponential from x times that exponential, then add C.
Used in: Integration by parts helps calculate work from product expressions for changing forces.

If your child says…“I add the extra integral.”

Ask: How did the extra exponential move across the equality?
What’s true: The extra derivative term must be subtracted to recover the required integrand.

Find a total between limits

Four and eight rectangles approximate the same triangular region beneath a rising line.

7. Definite Integral

Add signed contributions across an interval to find one total.

Show: Compare the diagram’s rectangles. Delta x labels their width.
Ask: What exact area do their totals approach?
Say: Add height times width for each rectangle. As widths shrink, the total approaches the triangle’s area: $\int_0^1 2x\,dx=1$ .
Used in: Integrating changing electrical power gives energy use, helping estimate electricity bills.

If your child says…“The answer still needs an unknown constant.”

Ask: How could adding different constants give the same fixed triangle’s area?
What’s true: The fixed region has one area, so its integral has no arbitrary constant.

Reflected triangular regions have equal areas and heights that add to one.

8. Some Properties of Definite Integrals

Use reflection and symmetry to simplify accumulated totals.

Show: Look at the diagram’s two shaded triangles and their dashed midpoint lines. Compare the marked heights at the same horizontal position.
Ask: What total will adding these equal, reflected areas give?
Say: Reflection moves the triangle without changing its area. The two graph heights add to one everywhere, so the areas total one. Each integral is therefore one half.
Used in: Symmetric shapes let designers calculate full areas from half, reducing repeated work.

If your child says…“For equal halves, one half’s integral gives the whole.”

Ask: How many equal triangles contribute to the square’s whole area?
What’s true: The two equal triangular areas add to the whole square’s area.

A real problem

A polar planimeter measured irregular areas by tracing their boundaries.

Jakob Amsler developed a practical polar planimeter in 1854.

Tracing a flat shape’s boundary let the instrument measure its area.

Planimeters evaluate definite integrals mechanically.

Science Museum Group Powerhouse Museum Virginia Commonwealth University

Paper map with an irregularly shaped lake on a wooden desk.

Concept Lab · Common misconception

The tempting mistake in Integrals

Open standalone misconception page →

Does the power integration rule work for 1/x?

Wrong

Integrating x⁻¹ gives x⁰/0 by the usual power rule.

Why it feels right. The power rule works for most familiar exponents.

Right

The power rule for antiderivatives excludes exponent −1 because the new denominator would be zero.

Worked example

Power rule attempt

x⁻¹→Exponent −1 + 1 = 0→Division by zero

Derivative check

ln(abs(x))→Differentiate for x ≠ 0→1/x

Adding 1 to exponent −1 would make the divisor zero. A derivative check gives the correct antiderivative on intervals away from zero.

Differentiate the natural log of the absolute value of x and you obtain 1/x.
On any interval excluding zero, add a constant to get the family of antiderivatives.

Ask your child eight quick questions.

One per step. The answer and the reason are given.

What remains undetermined when only the derivative is given?

The added constant. Different constants give the same derivative.
Which entire factor becomes dt here?

$2x\,dx$ Differentiating x squared supplies the factor twice x.
Which missing factor prevents direct substitution of sine into the power rule here?

$\cos x$ Cosine is the derivative of sine.
What coefficient outside inverse tangent corrects the first derivative shown?

$\frac12$ The uncorrected derivative is twice the required expression.
What operation joins the two expressions inside the combined logarithm?

Division. Multiplication would give a sum of logarithms.
Which sign belongs before the remaining integral in the last line?

Minus. Subtracting removes the extra term created by the product rule.
What is this exact integral, including any constant needed?

1, with no arbitrary constant. The triangle has a fixed base and height, so its area is fixed.
The two equal integrals together give one. What is each integral?

One half. One is the combined total of two equal areas.

15 more questions with answers

Teach Integrals in eight steps.

Why it matters